Modal Analysis

What Is It?

Modal analysis is the process of computing the natural frequencies and mode shapes of a structural system using finite element analysis. It solves the eigenvalue problem arising from the undamped, free-vibration equation of motion. The results — a set of frequencies and corresponding mode shapes — form the foundation for most subsequent dynamic analyses, including harmonic response, random vibration and modal transient analysis.

Why It Matters

Modal analysis is not a final engineering answer in itself. It is a diagnostic and foundational step. It tells the engineer what the structure's dynamic character is — where resonances exist, which modes are global or local, which directions have significant modal participation and whether the model itself is physically credible. Most frequency-domain dynamic analyses are built on the modal results. A poor modal model undermines everything derived from it.

The quality of a random vibration, harmonic response or transient modal result depends on the modal model underneath it. Invest time in getting the modal analysis right.

Model Preparation

A modal analysis is only as good as the finite element model it is based on. The key modelling decisions that affect modal results are mass representation, stiffness representation, constraints, connection stiffness, prestress effects and contact assumptions. Each of these must be considered carefully before running the eigenvalue extraction.

Mass Representation

Accurate mass representation is critical for modal analysis. Structural mass is usually well-represented by element mass matrices. Non-structural mass — equipment, fasteners, cables, payload, thermal protection — must be included, typically as lumped or distributed mass. The location of non-structural mass matters: mass placed at a modal antinode (point of maximum motion) has more effect on frequency than the same mass at a node (point of minimal motion). Rotary inertia of equipment should be included where rotational modes are relevant.

MODELLING CONSIDERATION: Omitting non-structural mass — cables, equipment, thermal protection, fasteners — can shift natural frequencies significantly. Include all mass that participates in the modes of interest.

Solver Methods

Several eigenvalue extraction methods are used in FEA solvers. Each has strengths in terms of efficiency, accuracy and the type of problems it handles well. The engineer does not usually need to implement these methods, but understanding their characteristics helps in selecting appropriate solver settings.

Selecting the Frequency Range

The frequency range for eigenvalue extraction should be guided by the excitation environment, not by an arbitrary default. If the excitation spectrum extends to 1000 Hz, modes below 1000 Hz are directly relevant and must be extracted. Extracting modes far above the excitation range adds computational cost without engineering value. However, some modes above the excitation range may still be needed for modal truncation adequacy in subsequent dynamic analyses.

COMMON MISTAKE: Extracting hundreds of modes without understanding the relevant excitation range. This is computationally expensive and makes it harder to identify the modes that actually matter.

Modal Truncation

In modal superposition methods — used for harmonic, random and transient analysis — the response is approximated by summing the contributions of a finite number of modes. Modes above the extraction cut-off are omitted. This truncation introduces error. The adequacy of the modal set can be assessed using effective modal mass: if the cumulative effective mass in the excitation direction approaches the total structural mass, the modal set is generally adequate. If significant mass remains unrepresented, more modes or residual mass correction may be needed.

Cumulative effective mass ratio (direction i):

Σ m_eff,n,i  /  M_total,i

where:
M_total,i  =  total structural mass in direction i

Aim for this ratio to approach 1.0 for the excitation directions of interest.

Effective Mass Table

The effective mass table is one of the most useful outputs of a modal analysis. It summarises which modes matter for excitation in each direction. The table below shows an illustrative format — the values shown are hypothetical for format demonstration only.

DYNAMIC CHECK: Confirm that the cumulative effective modal mass in each excitation direction adequately represents the total mass. If significant mass is missing, the modal basis may be insufficient for subsequent dynamic analysis.

Check the Mode Shapes, Not Just the Frequency Table

A frequency table lists numbers. It cannot reveal whether a mode is physically realistic, whether it represents a genuine structural mode or a modelling artefact, or whether a local mode is structurally significant. Visual inspection of mode shapes is essential. Animate each mode, identify the deformation type and confirm it makes engineering sense.

Check the mode shapes, not just the frequency table. An unrealistic mode shape reveals a modelling problem that no frequency table can expose.

Identifying Unrealistic Local Modes

Local modes — deformation confined to a small region — can be physically real or modelling artefacts. A panel mode on a thin skin may be real and significant. A mode confined to a single element or a poorly connected part is likely a modelling problem. Distinguish between them by checking the strain energy distribution and the effective mass. A mode with negligible effective mass and localised strain energy in a questionable region should be investigated.

• Check whether the mode involves a physically plausible deformation pattern
• Check effective mass — negligible mass may indicate a local or artefact mode
• Check strain energy distribution — concentrated energy in one element suggests a problem
• Check whether the mode disappears with mesh refinement — artefact modes are often mesh-dependent

Prestress Effects

If a structure carries static preload — bolt tension, pressure load, thermal stress, centrifugal loading — this can alter the stiffness and therefore the natural frequencies. Tensile prestress typically raises frequencies (stress stiffening); compressive prestress typically lowers them (stress softening). For rotating machinery, pressurised structures or preloaded bolted joints, prestress-modified modal analysis may be necessary. Ignoring prestress when it is significant can produce misleading frequency predictions.

Key Takeaways

• Modal analysis is a foundational step — most frequency-domain dynamic analyses depend on it
• Model preparation — mass, stiffness, constraints, joints — determines the quality of the modal results
• Select the frequency range based on the excitation environment, not an arbitrary default
• Check effective mass for modal truncation adequacy in subsequent dynamic analysis
• Always visually inspect mode shapes — a frequency table alone is insufficient verification